Question 1

The sales tax $T$ , in dollars, I have to pay when I buy D dollars worth of merchandise is giv- en by $T = 0.08 D$ .
If I decide to buy $35 more in merchandise, how much additional sales tax will I have to pay?
A) $2.8
B) $3.04
C) $0.6
D) $0.08

Accepted Answer

The additional sales tax for buying $35 more in merchandise is calculated by applying the sales tax rate of 0.08 to the additional amount: $0.08 \times 35 = 2.8$.

Question 2

The following table shows the running speed  , in feet per second, of animals of length L , in inches. Use the regression line for the data to estimate the running speed of animals that are 14 inches long. $$\begin{array} { | l | l | l | l | l | } 
\hline L & 3.5 & 6.3 & 9.4 & 9.8 \
\hline R & 8.22 & 15.78 & 24.02 & 24.93 \
\hline
\end{array}$$
A) 17.64 feet per second
B) 36.33 feet per second
C) 17.36 feet per second
D) 36.19 feet per second

Accepted Answer

The answer of The following table shows the running speed...

Question 3

The following table shows the depth D , in inches, of water in a pond after $d$ days of eva- poration. Find a linear model for depth as a function of days. $$\begin{array} { | l | l | l | l | l | } 
\hline d & 4 & 6 & 8 & 10 \
\hline D & 779 & 777 & 775 & 773 \
\hline
\end{array}$$
A) $D = - 2 d + 787$
B) $D = 783 d + - 1$
C) $D = - 1 d + 783$
D) $D = 787 d + - 2$

Accepted Answer

To find a linear model, we need to find the slope and y-intercept of the line that passes through the given points. Using the points (4, 779) and (10, 773), we get a slope of $(773-779)/(10-4)=-1$. Using the slope-intercept form $y = mx + b$ and one of the points, say (4, 779), we can solve for the y-intercept $b$: $779 = (-1)(4) + b$, which gives $b = 783$. Putting the slope and y-intercept together, we get $D = -1d + 783$. Therefore, the correct choice is C.

Question 4

Suppose $f$ is a linear function of $x$ . The slope is 2.44. An increase of 6 units in $x$ causes what change in $f$ ?
A) A decrease of 14.64 units
B) An increase of 14.64 units
C) An increase of 2.46 units
D) A decrease of 2.46 units

Accepted Answer

Since $f$ is a linear function of $x$, we can write its equation as:

$f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

The problem tells us that the slope is 2.44, so we have:

$f(x) = 2.44x + b$

To find out what happens to $f$ when $x$ increases by 6 units, we need to find the value of $f(x + 6)$. We have:

$f(x + 6) = 2.44(x + 6) + b$

We don't know the value of $b$, but we don't need it, because we only need to find the difference between $f(x + 6)$ and $f(x)$. We have:

$$\begin{aligned} f(x + 6) - f(x) &= (2.44(x + 6) + b) - (2.44x + b) \ &= 2.44x + 14.64 - 2.44x \ &= 14.64 \end{aligned}$$

Therefore, an increase of 6 units in $x$ causes an increase of 14.64 units in $f$. The answer is (B).

Question 5

The following table shows the gross national product G , in trillions of dollars,  years after 2005. Use linear regression to model G as a function of  . $$\begin{array} { | l | l | l | l | l | } 
\hline t & 0 & 1 & 3 & 5 \
\hline G & 12.72 & 13.52 & 14.27 & 14.65 \
\hline
\end{array}$$
A) $G = 12.96 t + 0.37$
B) $G = 0.37 t + 12.96$
C) $G = 0.37 t + 12.72$
D) $G = 0.8 t + 12.72$

Accepted Answer

The answer of The following table shows the gross national...

Question 6

The following table shows the running speed  , in centimeters per second, of ants as a func- tion of the temperature $T$ in degrees Celsius. Based on the regression equation, how fast would an ant run
When the temperature is 30.9 degrees Celsius? $$\begin{array} { | l | l | l | l | l | } 
\hline T & 25.6 & 30.3 & 32.3 & 34.8 \
\hline R & 2.46 & 3.96 & 4.14 & 4.75 \
\hline
\end{array}$$
A) 4.4 centimeters per second
B) 4.98 centimeters per second
C) 4.3 centimeters per second
D) 3.99 centimeters per second

Accepted Answer

The answer of The following table shows the running speed...

Question 7

Which of the tables below show linear data? Table 1 $$\begin{array}{l}
\begin{array} { | l | l | l | l | l | } 
\hline x & 2 & 4 & 6 & 8 \
\hline \mathrm { y } & 36.07 & 39.39 & 43.45 & 46.89 \
\hline
\end{array}\
\text { Table } 2\
\begin{array} { | l | l | l | l | l | } 
\hline x & 2 & 4 & 6 & 8 \
\hline \mathrm { y } & 35.82 & 39.38 & 42.94 & 46.5 \
\hline
\end{array}
\end{array}$$
A) Table 1 only
B) Table 2 only
C) Both Table 1 and Table 2
D) Neither table is linear.

Accepted Answer

In Table 2, the data points form a straight line when plotted on a graph, indicating that the relationship between x and y is linear. In Table 1, the data points do not form a straight line when plotted on a graph, indicating a non-linear relationship.

Question 8

Solve the equation $$7 x + 3 y - 7 = 0$$ for $y$ . 
A) $y = \frac { 7 - 7 x } { 3 }$ or $y = 2.33 - 2.33 x$

B) $y = \frac { 7 + 7 x } { 3 }$ or $y = 2.33 + 2.33 x$

C) $y = \frac { 7 x - 7 } { 3 }$ or $y = 2.33 x - 7$

D) None of the above.

Accepted Answer

First, isolate the term with $y$ by subtracting 7 and then dividing by 3: $$7x + 3y - 7 = 0 \implies 3y = -7x + 7 \implies y = \frac{7-7x}{3}$$ This is the only option in A, so it must be the answer. The second option in A, $y = 2.33 - 2.33x$, is equivalent to the answer when simplified, so it is also correct. Option B is incorrect because it has a positive sign in front of the $7x$ when it should be negative to match the given equation. Option C is also incorrect because it has a subtraction of 7 in the numerator instead of the denominator, which changes the value of y.

Question 9

Solve the equation $10 x - 5 y = 6$ for $y$ . 
A) $y = \frac { 6 - 10 x } { 5 }$ or $y = 1.2 - 2 x$

B) $y = \frac { 6 + 10 x } { 5 }$ or $y = 1.2 + 2 x$

C) $y = \frac { 10 x - 6 } { 5 }$ or $y = 2 x - 1.2$

D) None of the above.

Accepted Answer

To solve for y, we need to isolate y on one side of the equation.

Starting with $10x - 5y = 6$, we can first subtract $10x$ from both sides:

$$-5y = 6 - 10x$$

Next, we can divide both sides by $-5$ to solve for $y$:

$$y = \frac{10x - 6}{5}$$

So our answer is choice C. It can also be written as $y = 2x - 1.2$.

Question 10

Find a linear model for the following data table. $$\begin{array} { | l | l | l | l | l | } 
\hline x & 3 & 6 & 9 & 12 \
\hline y & 6 & 12 & 18 & 24 \
\hline
\end{array}$$
A) $y = 2 x + 0$
B) $y = 0 x + 2$
C) $y = 6 x + - 12$
D) $y = - 12 x + 6$

Accepted Answer

The data shows a consistent increase; for every increase of 3 in $x$, $y$ increases by 6. This indicates a slope ($m$) of $2$, and since the line passes through the origin, the y-intercept ($b$) is $0$. Thus, the linear model is $y = 2x + 0$.

Question 11

The following table shows the running speed  , in centimeters per second, of ants as a func- tion of the temperature $T$ in degrees Celsius. Use linear regression to model running speed as a function of
Temperature. $$\begin{array} { | l | l | l | l | l | } 
\hline T & 25.6 & 30.3 & 32.3 & 34.8 \
\hline R & 2.45 & 3.34 & 4.57 & 4.9 \
\hline
\end{array}$$
A) $R = 0.19 T + 2.45$
B) $T = 0.19 R + 2.45$
C) $R = 0.28 T - 4.84$
D) $T = 0.28 R - 4.84$

Accepted Answer

To use linear regression, we need to find the slope and the y-intercept of the line of best fit. Using a graphing calculator or statistical software, we can find that the slope is approximately 0.28 and the y-intercept is approximately -4.84. Therefore, the equation of the line of best fit is R = 0.28T - 4.84. Therefore, the best choice is C.

Question 12

Solve the equation $11.59 x - 8.34 y = 11.51$ for $y$ . 
A) $y = \frac { 11.51 - 11.59 x } { 8.34 }$ or $y = 1.38 - 1.39 x$

B) $y = \frac { 11.51 + 11.59 x } { 8.34 }$ or $y = 1.38 + 1.39 x$

C) $y = \frac { 11.59 x - 11.51 } { 8.34 }$ or $y = 1.39 x - 1.38$

D) None of the above.

Accepted Answer

We want to isolate y on one side of the equation, so we begin by subtracting 11.59x from both sides:

\begin{align*}
11.59 x - 8.34 y &= 11.51 \
-8.34 y &= 11.51 - 11.59 x \
y &= \frac{11.59 x - 11.51}{8.34} \
\end{align*}

This gives us the first part of option C. To check if the second part is correct, we can simplify the expression:

\begin{align*}
y &= \frac{11.59 x - 11.51}{8.34} \
&= \frac{1.39(8.34)x - 1.38(8.34)}{8.34} \
&= 1.39 x - 1.38
\end{align*}

So both parts of option C are correct, and it is the best choice.

Question 13

Suppose $f$ is a linear function with $f ( 8 ) = 27.92$ and $f ( 11 ) = 35.39$ . Find an equation for $f$ .
A) $f ( x ) = 0.4 x + 27.92$
B) $f ( x ) = 27.92 x + 0.4$
C) $f ( x ) = 2.49 x + 8$
D) $f ( x ) = 8 x + 2.49$

Accepted Answer

Since $f$ is a linear function, we can use the slope-intercept form of a linear equation, which is $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. From the given information, we can find the slope as $\frac{35.39-27.92}{11-8} = 2.49$ and the y-intercept as $\left(0, f(0)\right)$ where $f$ is the linear function. Since $f$ is linear, we know that the y-intercept will be the same as $f(0)$ and so we need to find the value of $f(0)$. To do this, we can use the point-slope form of a linear equation which is $(y-y_1) = m(x-x_1)$ where $(x_1,y_1)$ is a point on the line and $m$ is the slope. We can use $(8,27.92)$ as our point and $m = 2.49$ and then solve for $y$ to get $f(x) = 2.49x + 8$.

Question 14

Solve the system of equations $$\begin{array} { l } 
6.52 x + 15.84 y - 5.88 = 0 \
5.72 x = 7.73 - 15.42 y
\end{array}$$ 
A) $x = - 3.2$ and $y = 1.69$
B) $x = 1.69$ and $y = - 3.2$
C) $x = - 2.77$ and $y = 2.63$
D) $x = 2.63$ and $y = - 2.77$

Accepted Answer

The correct answer is A because when you solve the system of equations, you get x = -3.2 and y = 1.69.

Question 15

The following table shows the money M left in your cookie jar after $\mathrm { W }$ withdrawals from the jar. Find a linear model for M in terms of W. $$\begin{array} { | l | l | l | l | l | } 
\hline x & 3 & 4 & 5 & 6 \
\hline y & 136 & 121 & 106 & 91 \
\hline
\end{array}$$
A) $M = 136 w - 15$
B) $M = 181 w - 15$
C) $M = - 15 w + 136$
D) $M = - 15 w + 181$

Accepted Answer

The answer of The following table shows the money M...

Question 16

The following table shows the number C , in millions, of cars sold in the U.S.  years after 2005. Use the regression line for the data to estimate the number of cars sold in 2013.

A) $3.37$ million
B) $4.51$ million
C) $3.16$ million
D) $4.11$ million

Accepted Answer

Using the regression line for the data, the value of C for t=8 (2013−2005=8) can be estimated as:

C=−0.80×8+10.63=4.03

Therefore, the estimated number of cars sold in 2013 is 4.03 million, which is closest to choice D (4.11 million).

Question 17

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity $V$ , in feet per second, after  seconds is given by $V = 32 t + 8$ .
By how much does velocity increase over a 6-second period?
A) 32.14 feet per second
B) 192.51 feet per second
C) 32 feet per second
D) 192 feet per second

Accepted Answer

The answer of If a rock is thrown downward with...

Question 18

We have $18.33 to spend on vegetable plants for our garden. Tomato plants cost $0.87 each, and pepper plants cost $0.33 each. There is space in the garden for 31 plants.
How many of each are bought?
A) 13 tomatoes and 18 peppers
B) 16 tomatoes and 15 peppers
C) 18 tomatoes and 13 peppers
D) 15 tomatoes and 16 peppers

Accepted Answer

To solve, set up the equations: 0.87T + 0.33P = 18.33 and T + P = 31. Solving these, T = 15 and P = 16.

Question 19

Find a linear model for the following data table. Round the slope to three decimal places. $$\begin{array} { | l | l | l | l | l | } 
\hline x & 3 & 4.82 & 6.64 & 8.46 \
\hline y & 6 & 12.56 & 19.12 & 25.68 \
\hline
\end{array}$$
A) $y = 3.604 x + - 4.81$
B) $y = - 4.813 x + 3.6$
C) $y = 3.604 x + 6$
D) $y = 6 x + 3.6$

Accepted Answer

To find the linear model, we can use two points from the table to calculate the slope (m) of the line. Using the points (3, 6) and (8.46, 25.68):$$m = \frac{25.68 - 6}{8.46 - 3} = \frac{19.68}{5.46} \approx 3.604$$Next, we use the slope and one point to find the y-intercept (b) using the formula $y = mx + b$. Using the point (3, 6):$$6 = 3.604 \cdot 3 + b$$$$b = 6 - 3.604 \cdot 3 \approx -4.812$$Thus, the linear model is $y = 3.604x - 4.81$, which matches choice A.

Question 20

Suppose $f$ is a linear function with $f ( 8 ) = 26.2$ . If the slope of $f$ is 2.4, what value of $x$ gives $f ( x ) = 12$ ?
A) $x =$36.46 
B) $x =$2.08  
C) $x =$35.8 
D) $x =$2.98

Accepted Answer

Since $f$ is a linear function, we can find its equation using the slope-intercept form: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
We know that the slope of $f$ is 2.4, so we have $f(x) = 2.4x + b$. To find the value of $b$, we use the fact that $f(8) = 26.2$:
$$f(8) = 2.4(8) + b = 26.2$$
$$b = 26.2 - 19.2 = 7$$
So now we can write the equation of $f$ as $f(x) = 2.4x + 7$.
To find the value of $x$ that gives $f(x) = 12$, we set $f(x) = 12$ and solve for $x$:
$$2.4x + 7 = 12$$
$$2.4x = 5$$
$$x = \frac{5}{2.4}\approx2.08$$
Therefore, the answer is B.

Quiz 3: Straight Lines and Linear Functions

The sales tax $T$ , in dollars, I have to pay when I buy D dollars worth of merchandise is giv- en by $T = 0.08 D$ . If I decide to buy $35 more in merchandise, how much additional sales tax will I have to pay?

The following table shows the depth D , in inches, of water in a pond after $d$ days of eva- poration. Find a linear model for depth as a function of days. $\begin{array} { | l | l | l | l | l | } \hline d & 4 & 6 & 8 & 10 \\\hline D & 779 & 777 & 775 & 773 \\\hline\end{array}$

Suppose $f$ is a linear function of $x$ . The slope is 2.44. An increase of 6 units in $x$ causes what change in $f$ ?

Solve the equation $7 x + 3 y - 7 = 0$ for $y$ .

Solve the equation $10 x - 5 y = 6$ for $y$ .

Find a linear model for the following data table. $\begin{array} { | l | l | l | l | l | } \hline x & 3 & 6 & 9 & 12 \\\hline y & 6 & 12 & 18 & 24 \\\hline\end{array}$

Solve the equation $11.59 x - 8.34 y = 11.51$ for $y$ .

Suppose $f$ is a linear function with $f ( 8 ) = 27.92$ and $f ( 11 ) = 35.39$ . Find an equation for $f$ .

Solve the system of equations $\begin{array} { l } 6.52 x + 15.84 y - 5.88 = 0 \\5.72 x = 7.73 - 15.42 y\end{array}$

The following table shows the money M left in your cookie jar after $\mathrm { W }$ withdrawals from the jar. Find a linear model for M in terms of W. $\begin{array} { | l | l | l | l | l | } \hline x & 3 & 4 & 5 & 6 \\\hline y & 136 & 121 & 106 & 91 \\\hline\end{array}$

The following table shows the number C , in millions, of cars sold in the U.S. years after 2005. Use the regression line for the data to estimate the number of cars sold in 2013.

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity $V$ , in feet per second, after seconds is given by $V = 32 t + 8$ . By how much does velocity increase over a 6-second period?

We have $18.33 to spend on vegetable plants for our garden. Tomato plants cost $0.87 each, and pepper plants cost $0.33 each. There is space in the garden for 31 plants. How many of each are bought?

Find a linear model for the following data table. Round the slope to three decimal places. $\begin{array} { | l | l | l | l | l | } \hline x & 3 & 4.82 & 6.64 & 8.46 \\\hline y & 6 & 12.56 & 19.12 & 25.68 \\\hline\end{array}$

Suppose $f$ is a linear function with $f ( 8 ) = 26.2$ . If the slope of $f$ is 2.4, what value of $x$ gives $f ( x ) = 12$ ?

Quiz 3: Straight Lines and Linear Functions

The sales tax TTT , in dollars, I have to pay when I buy D dollars worth of merchandise is giv- en by T=0.08DT = 0.08 DT=0.08D . If I decide to buy $35 more in merchandise, how much additional sales tax will I have to pay?

Suppose fff is a linear function of xxx . The slope is 2.44. An increase of 6 units in xxx causes what change in fff ?

Solve the equation 7x+3y−7=07 x + 3 y - 7 = 07x+3y−7=0 for yyy .

Solve the equation 10x−5y=610 x - 5 y = 610x−5y=6 for yyy .

Find a linear model for the following data table. x36912y6121824\begin{array} { | l | l | l | l | l | } \hline x & 3 & 6 & 9 & 12 \\\hline y & 6 & 12 & 18 & 24 \\\hline\end{array}xy​36​612​918​1224​​

Solve the equation 11.59x−8.34y=11.5111.59 x - 8.34 y = 11.5111.59x−8.34y=11.51 for yyy .

Suppose fff is a linear function with f(8) =27.92f ( 8 ) = 27.92f(8) =27.92 and f(11) =35.39f ( 11 ) = 35.39f(11) =35.39 . Find an equation for fff .

Solve the system of equations 6.52x+15.84y−5.88=05.72x=7.73−15.42y\begin{array} { l } 6.52 x + 15.84 y - 5.88 = 0 \\5.72 x = 7.73 - 15.42 y\end{array}6.52x+15.84y−5.88=05.72x=7.73−15.42y​

The following table shows the number C , in millions, of cars sold in the U.S. years after 2005. Use the regression line for the data to estimate the number of cars sold in 2013.

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity VVV , in feet per second, after seconds is given by V=32t+8V = 32 t + 8V=32t+8 . By how much does velocity increase over a 6-second period?

We have $18.33 to spend on vegetable plants for our garden. Tomato plants cost $0.87 each, and pepper plants cost $0.33 each. There is space in the garden for 31 plants. How many of each are bought?

Suppose fff is a linear function with f(8) =26.2f ( 8 ) = 26.2f(8) =26.2 . If the slope of fff is 2.4, what value of xxx gives f(x) =12f ( x ) = 12f(x) =12 ?

The sales tax $T$ , in dollars, I have to pay when I buy D dollars worth of merchandise is giv- en by $T = 0.08 D$ . If I decide to buy $35 more in merchandise, how much additional sales tax will I have to pay?

Suppose $f$ is a linear function of $x$ . The slope is 2.44. An increase of 6 units in $x$ causes what change in $f$ ?

Solve the equation $7 x + 3 y - 7 = 0$ for $y$ .

Solve the equation $10 x - 5 y = 6$ for $y$ .

Find a linear model for the following data table. $\begin{array} { | l | l | l | l | l | } \hline x & 3 & 6 & 9 & 12 \\\hline y & 6 & 12 & 18 & 24 \\\hline\end{array}$

Solve the equation $11.59 x - 8.34 y = 11.51$ for $y$ .

Suppose $f$ is a linear function with $f ( 8 ) = 27.92$ and $f ( 11 ) = 35.39$ . Find an equation for $f$ .

Solve the system of equations $\begin{array} { l } 6.52 x + 15.84 y - 5.88 = 0 \\5.72 x = 7.73 - 15.42 y\end{array}$

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity $V$ , in feet per second, after seconds is given by $V = 32 t + 8$ . By how much does velocity increase over a 6-second period?

Suppose $f$ is a linear function with $f ( 8 ) = 26.2$ . If the slope of $f$ is 2.4, what value of $x$ gives $f ( x ) = 12$ ?